Topological Protection in a Strongly Nonlinear Interface Lattice

TL;DR

This paper investigates a strongly nonlinear one-dimensional mechanical lattice, revealing stable topologically insulated modes parameterized by energy, and introduces a method to predict their existence using geometric Zak Phase calculations.

Contribution

It demonstrates the existence and stability of nonlinear topological modes in a strongly nonlinear lattice and links their excitability to the Zak Phase, providing a new predictive approach.

Findings

Nonlinear topological modes are stable within certain energy ranges.

Zak Phase can predict the energy threshold for mode excitability.

Numerical simulations validate the theoretical predictions.

Abstract

Mechanical topological insulators are well understood for linear and weakly nonlinear systems, however traditional analysis methods break down for strongly nonlinear systems since linear methods can not be applied in that case. We study one such system in the form of a one-dimensional mechanical analog of the Su-Schrieffer-Heeger interface model with strong nonlinearity of the cubic form. The frequency-energy dependence of the nonlinear bulk modes and topologically insulated mode is explored using Numerical continuation of the system's nonlinear normal modes (NNMs), and the linear stability of the NNMs are investigated using Floquet Multipliers (FMs) and Krein signature analysis. We find that the nonlinear topological lattice supports a family of topologically insulated NNMs that are parameterized by the total energy of the system and are stable within a range of frequencies. Next, it is shown that empirical calculations of the geometric Zak Phase can define an energy threshold to predict the excitability of the nonlinear topological mode, and that this threshold coincides with the energy that the topological NNM intersects the linear bulk-spectrum. These predictions are validated with numerical simulations of the nonlinear topological system. These results are also tested for parametric perturbations which preserve and break chirality in the system. Thus, we provide a new method for analyzing and predicting the existence of topologically insulated modes in a strongly nonlinear lattice based on the physical observable of band topology.